The Boson Normal Ordering Problem and Generalized Bell Numbers

For any function F(x) having a Taylor expansion we solve the bosonnormal ordering problem for $F [(a^dag)^r a^s]$, with r, s positive integers, $F [(a, a^dag]=1$, i.e., we provide exact and explicitexpressions for its normal form $mathcal{N} {F [(a^dag)^r a^s]} = F [(a^dag)^r a^s]$, wherein $ mathcal{N} (F) $ all a's are to theright. The solution involves integer sequences of numbers which, for $ r, s geq 1 $, are generalizations of the conventional Bell and Stirling numbers whose values they assume for $ r=s=1 $. A completetheory of such generalized combinatorial numbers is given including closed-form expressions(extended Dobinski-type formulas), recursion relations and generating functions. These last arespecial expectation values in boson coherent states.AMS Subject Classification: 81R05, 81R15, 81R30, 47N50.